#153265

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(6,7)(9,10,16,23)(11,17,22,15)(12,18,13,33)(14,36,37,27)(19,34,20,29)(21,31,40,30)(24,32,28,35)(25,38,26,39);(5,6)(7,8)(9,11)(10,17)(12,28)(13,24)(14,21)(15,23)(16,22)(18,35)(19,38)(20,39)(25,29)(26,34)(27,30)(31,36)(32,33)(37,40);(2,4)(5,7,8,6)(10,18)(11,24)(12,29)(13,34)(14,30)(15,17)(19,21)(20,40)(22,28)(23,33)(25,36)(26,27)(31,37)(32,38)(35,39)

Polycyclic presentation:

2^9|a^2=i,c^2=d,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,e^b=eg,f^b=fg,g^b=gh,h^b=hi,e^c=eg,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60461);